The student is absolutely correct. I don't think it's even open for debate. Cutting anything in half requires exactly one cut; cutting in thirds requires two. It's as simple as that. The teacher that crafted the question, or worse yet, the publisher of a textbook that may have provided the test question, needs to take a hard look at whether or not they are in the correct profession.
The fact that the teacher not only marked the answer wrong (which could have just resulted from looking at a publisher-provided answer key) but actually wrote down a completely incorrect justification for the teacher's incorrect answer is rather disturbing to me. Also, this did not occur in a vacuum. Either no other students answered the question correctly, or the teacher saw the question being answered correctly by others and repeatedly marked it wrong with the same justification. Either way, it causes concern about the teacher.
The fact that the teacher not only marked the answer wrong (which could have just resulted from looking at a publisher-provided answer key) but actually wrote down a completely incorrect justification for the teacher's incorrect answer is rather disturbing to me. Also, this did not occur in a vacuum. Either no other students answered the question correctly, or the teacher saw the question being answered correctly by others and repeatedly marked it wrong with the same justification. Either way, it causes concern about the teacher.